N = 土 立 / a + 2 Q O
-A
N みかけの縞本数
図20 商 内 変位に よ る み か け の縞本数
唱2・4qo
松井 ・ 加 藤 ・ 吉 川 ・ 田 代 : ホ ロ グ ラ フ ィ 干縞 の 局 在場所 に 注 目 し た 変位量の 測定 ( 回 転変位 と 面 内 変位の 測 定 )
図21 面内変位に よ る 縞 の傾 き 角
4. 結 論
隔間隔一縞間
一の 縞 二
円
一一ス 一タ ゾ 一ユ
一)
イ 7J
7一プ一←
。，n
a
t
11
T 一M
''1a十h、、
n u 一一。，
a 物 体 と ホロ グラム と の 距 離
ホログラフ ィ 干渉法における， 複合変位をする応用実験において， 真の垂直変位量と面内変位量を 求めるのに役立つ
このモデル実験により得られた ことを示す。
1 . 局在点の変化は， 回転量を一定とすれば， 面内変位量μ mを与えると， 局在場所はc mで測定で、き一 種の拡大機構となる。
2 . 局在場所んとすれば， Plは式(10) で与えられる。
3 . 縞の増減による記録は スポ ッ ト再生により可能となる。
4 . 縞の増減， 傾き角は， 式(14)， 式(15)で与えられる。
5 . 図2の変位方向ω)の真の面内変位量， およ び団転量は， 式 (10)と式 (14)の二つの式の連立方程 式を解くことにより， 求められる。
6 . 局在場所の測定方法は， なお思考を用するが， 局在場所の測定誤差による， 真の変位量 におよ ぶ誤差は， ムT= 28，ムPl，ムθ=( 2(/2 /T)ムん となり， 面内変位量に大きく影響する。 従が っ て， 面内変 位量の大きな引張 問題では良好な感度を有すると判 断される。
終りに本研究をすすめるに あた っ て， 理化学研究所の 山 口一郎氏と工業計測講座の研究室の 人達に より誠心援助を頂だいたことを深く感謝します。
参 考 文 献 本論文 は 昭和52 年度， 精機学会春季大会 に 発 表 した も の の 一 部で あ る 。
32
-1 ) J. LEROY: Localisation des franges en int erf eromet rie holog raphiq : NOUV ELLE REVUED ' opt iqu vol 6 NO. 6 ( 1 975 )
2 ) 山口 一郎 : レ ー ザー を 用 い た変 形 の 測 定: 画像技術， 3 月 号， 1976年
3 ) 辻内順平， 武 谷 直 谷， 松 田 浄 史 : ホ ロ グ ラ フ ィ に よ る 変 形 の 測 定: 応 用 物理37巻 第 9 号 ( 1 968 ) 4 ) M . ボ ル ン ， E. ウ ォ ル フ : 光 学の 原理 II， III 東 海 大出 版 会
The Measurment of Displacement by Fringe-Iocalization in Holographic Interferometry .
Hiroshi MATS UI Kazuo YOSHIKA WA
Tadashi KA TO Hatsuzou TASHIRO
This paper deals with the basic reseach of material deformation by holographic inter.
ferometry.
The authors paid attention to the fact that fringe-localization and fringe pattern were vari
ed by a combination of displacement (rotation and translation).
For this purpose， the theories of fringe-localization and fringe pattern are analized， and sim
plefied models are used.
As a result， the measurment of each component quantity is made possible.
( 1977 年 10 月 2 0 日 受理 )
Analysis of Diffraction Pattern by Wedge Shape and its Application to Angle Measurement
Norio ITOH and Tadashi KATOH
Since gas laser was put to use, its applied researches in preCisiOn measurement have been actively made, taking advantage of coherence which is its greatest characteristic. The present paper, using elementary function, shows that Fraunhofer diffraction by wedge-shaped aperture and wedge-shaped opaque forms diffraction pattern or hyperbola group, and then describes the method of wedge-shaped angle measurement.
1 . Introduction
The method of angle measurement which has been done up to this time could be classi
fied into the following two methods according to the shapes forming the angle. One is the mehod of direct angle measurement like the protractor. It includes angle reading device
·which is attached to the measuring microscope or the profile projector. The other is the indirect method like the angle interferometer.
As the wedge-shaped angle is so small and no other instrument except the microscope and projector can measure it, the authors experimented on the indirect method to measure such wedge-shaped angle. For that purpose, we first analyze the distribution of intensity of diffraction pattern and verify that the dark and light stripes are hyperbolas and from these we propose two simple methods.
Using similar principle, a few studies1J-3J on the measurement of displacement and profile have been done already, but no examples could be found applying it to the angle measure
ment.
2. Diffraction pattern by wedge-shaped aperture
The wedge-shaped aperture, which has constant angle a, is assumed to be sufficiently long and without thickness. As shown in figure 1, let the vertex of wedge be the origin.
Axis �stands for the bisector of angle a and axis 17 crosses it at right angles. Then, cor
responding with them, cnosider the axes of x- y co-ordinate on the screen which is situated far away from wedge. When monochromic plane wave of wavelength A is projected perpen
dicularly on the position where the width of wedge is w, the figure appearing on the screen could be approximately regarded as the Faunhofer diffraction pattern on the assumption that A { w{ R (R is the distance between wedge and screen). Therefore, its intensity I is
- 34
-(
sin¢)
2 n .I = I o --;;;- , ct>=AsmB (1) where /0 is the intensity in the direction of B = 0. Since ct> equals nn ( n= ± 1, ± 2, ··· ), the position of the center of dark fringe in diffraction pattern becomes
wsinB= n.t (n= ±1, ±2,, ... ) (2) Within the range where B is small enough, the equation sinB = y/R is approximately available. So the equation finally results
wy= n.tR (n= ±1, ±2, .. ·) (3) Then, assuming that the width of wedge is changed according to the equation 77 =m�
(m is the inclination of the wedge side), w equals 2n, and on the screen �=x could be utilized. Therefore the equation (3) becomes
n,tR
xy= Zm (n= ±1, ±2, .. ·) (4)
y
R
Wedge shaped aperture
Fig. 1. Diffraction pattern by wedge shaped aperture.
The right side of this equation is obviously constant, so that the diffraction pattern on the screen forms a groups of 2 n hyperbolas whose asymtotes are the axes of x- y co-ordinate.
3. Diffraction pattern by wdge-shaped opaque
Considering the wedge-shaped opaque as shown in figure 2, the diffraction figure by the wedge-shaped opaque has perfectly complementary relation with that of the wedge-shaped aperture. And from Babinet principle, Fraunhofer diffraction by this opaque shows the same intensity distribution as in the case of aper
ture on the screen except for the central part. Therefore in the case of the opaque, equation (4) obtained in the preceding sec
tion should be held as it is. If the vertical angle of wedge is equal, this opaque forms the same hyperbolas group as aperture.
Fig. 2. Schematic diagram of complementary relation between wedge shaped aperture and wedge shaped opaque.
4. Experimental analysis
It has been derived analytically that Fraunhofer diffraction pattern by wedge-shaped
N. Itoh. T Katoh
Analysis of Diffraction Pattern by Wedge Shape and its Application to Angle Measurement
aperture and opaque, both of which have the same angle, shows the same distribution of intensity except for its central part. In this section, we shall confirm it experimentally.
Since it is difficult to manufacture a pair of wedge-shaped aperture and opaque which have a complementary relation with one another, we confirm it indirectly, making use of aperture and opaque specimens with parallel sides. To set about this experiment, manufac·
ture some specimens with width w which should be similar to the measured portion of wedge-shaped specimen in size. Then there is no need to consider the complementary relation bet
ween them (see figure 3). Maintaining con
stant distance between specimen and screen, diffraction patterns are made. Measure the distance Yn between the center of the ob
tained pattern and dark fringe of n-th order.
In consequence, the relation between w and Yn is illustrated in figure 4. As it is evident in figure 4, the measured value Yn at the same order lay almost on the same hyper
bola in both cases of aperture and opaque.
Table 1 substantiates it further-more, where Wa.o is the width of each specimen measured
by profile projector. E
The sigh ,land R denote the wave length E of laser light and the distance between 20 specimen and screen respectively. The pro
duct ,tR denotes a constant in equation (3), � and Wa.bYnl n agrees with A.R in value. Hence, if the specimens of aperture and opaque I 0 are parallel and of the same width, those diffraction patterns have the same fringe space. And if the width changes, it gives a hyperbola in each order. The facts des-cribed above were confirmed experimentally within the experimental errors. Therefore, it is quite all right to consider that the
0
Aperture Opaque
Ill
-l -1
1-Fig.3. Specimens of aperture and opaque with two parallel sides for experimental analysis.
0.2 0.4 0.6
o Wa
• Wb
Wa,b (mm)
Fig.4. Fringe space Yn as a function of parallel slit width Wa.b for specimens of Fig. 3.
same result will be obtained in the case of wedge-shaped whose width changes continually.
Table 1. The calculated values, showing that fringe distance Yn at same order lay on the same hyperbola in both cases of aperture and opaque.
n 1 2 3 4 5 6
Cn 0.904 1.810 2.692 3.608 4.507 5.408 Wedge Aperture
Cn/n 0.904 0.905 0.897 0.902 0.901 0.901
shape Cn 0.917 1.778 2.662 3.527 4.439 5.299
Opaque
Cn/n 0.917 0.889 0.887 0.882 0.888 0.883 Notes: Cn=Wa.bYn. Cn!n=A.R
-36
-7 6.309 0.901 6.141 0.877
Nowadays, as we enter the save-energy and resources era, the effort to make various ma
chine parts small and fine is advanced and the manufactured goods come to be checked not only on the accuracy of their dimension but also that of profile. However, measurements of such small-sized parts still depend on a profile projector and a measuring microscope.
Applying the fact that diffraction pattern by wedge shape forms hyperbola, we propose two methods to measure the angle which seems to be one of the significant factor for the pro
file measurement of small aperture and small opaque having polygonal shape.
5. 1 Measurement using fringe spaces See figure 5, where A and B are the points plotted on axis � , and Wa and wb are the widths of wedge at those spots. Assum
ing that the length of AB is �ab, tan (a /2) equals (wa- wb)/2�ab· Letting Ya and Yb on axis y correspond to width Wa and wb, equa
tion below could be derived from equation (4).
tan!!..=nA.R2 2�ab Ya Yb
(
_!__l.)
(n=±1,±2,···) (5)Therefore, measuring values of �ab, Ya Fig.5. How to calculate angle, using fringe spaces.
and Yb on the screen, makes it possible to
obtain the value of the angle of wedge-shaped aperture and opaque. In addition as far as the values of Ya and Yb are concerned, 8 has to be
suffi-ciently small, on the other hand �ab is arditrary. y 5.2 Measurement using hyperbola
See figure 6, where p n (p, Cn/ p) is the point where hyperbola xy = Cn ( Cn= constant) crosses line x = P. The tangent line pass
ing the point p n is given by the equation y = -( Cn /P2) x + 2 Cn/ p, which crosses axis x at (2p, 0). So it is independent of the value of Cn. In other words, letting Ph P2,
····, Pn be the points where the perpendicular line passing given point p on axis x crosses each hyperbola, tangent lines passing Pt. P2,
····, P n cross a single point point Q on axis
x. Therefore OL---:l-( -P p,O)
--,--
---"'
Q ':--(2p,O ---:-
)---
X .Fig.6. How to calculate angle, using hyperbola.
N. Itoh, T. Katoh
Analysis of Diffraction Pattern by Wedge Shape and its Application to Angle Measurement
x = PQ p, Yn Using equating (4) in the same way,
tan�= nJ...R (n = ±1, ±2, ... ) (6) 2 2XYn
Like above, the wedge angle could be obtained by measuring PQ and PP n on the diffraction pattern.
5.3 Distinctions and controversial points
Making use of the alteration of fringe space and diffraction pattern by hyperbolas, these two methods described above show their distinctions for the comparatively small angles (e.g.
0-100 ). That is to say, diffraction patterns formed by using the portion of the wedge where width w is small and at the same time using the vertex of the wedge and the portion around it are needed. Especially when the hyperbolas are put to use, these methods are applied to the angle containing its vertex because such construction as to draw tangent lines are re
quired.
On the other hand, as a controvertial point for practical purposes, diffraction pattern obtained by directly projecting a fine laser beam into specimen has long length in the direc
tion being perpendicular to wedge side (namely the direction of axis y ), and is narrow in the direction of axis x as shown in figure 7. Therefore, those hyperbolas haven't sufficient dimensions to measure the fringe spaces and to draw tangent lines. In order to bring it into use, with the method using fringe spaces, the cylindrical lens should be placed in front of the specimen to magnify the diffraction pattern in the direction of axis x, so that the value of �ab in equation (5) will be enlarged to its suitable quantity. In the method using hyper
bolas, placing a convex lens behind the specimen to magnify the whole diffraction pattern
(a) (b)
Fig.7. Diffractograms of wedge shape (a) without lens.
(b) with cylindrical lens in front of specimen.
(c) with convex lens behind specimen.
- 38
-(c)
6. Summary
(1) In both cases of wedge-shaped aperture and opaque, the Fraunhofer diffraction pat
terns consisted of hyperbolas group. And it was confirmed experimentally.
(2) Two new methods of angle measurement making use of the result of (1) were pro
posed, and its distinctions and its controvertional points for practical purposes were descri · bed.
References
1) T.R.Pryor. O.L.Hageniers. and W.P.T.North : Appt. Opt., 11 (2), 308 (1972).
2) T.R.Pryor. O.L. Hageniers. and W.P.T. North Appl. Opt., 11 (2). 314 (1972).
3) O.L.Hageniers, T.R.Pryor, ·and W.P.T. North : Rev. Scient. Instr., 43 (11). 1688 (1972)
( 1977i!'-10 Jl20 B "ltlll!.)
非線形関数発生器と演算増幅器の特性について
明石 ー米 ・中川孝之・高瀬博文
1. は し が. き
油圧機構を含む機械系(実際の装置による)振動の実験結果にもとずいて その動作を解析し， あと にのべる動作方程式を得た。 この場合， 実際の装置における動作は， その動作の再現性がきわめて乏 しいので， アナログ計算機による動作と実際の装置の動作の相似性が成り立つと仮定し， 両者の動作 の比較検討を行なって， 油圧を用いた機械系の動作の解析することを目的とした。
これまで， この動作に対する取扱いは， 摩擦特性を数本の折れ線で表示し， ある動作が限られた範 囲内で線形動作をするものと仮定し， この範囲外では別の線形動作をするものとして解を求め， それ ぞ、れの線形動作を接ぎ合せて考察したものが多いi) そして一般に， このような非線形微分方程式であ らわされる動作を解析的に解くことが困難なので， 筆者らはアナコンを用いてこの動作の解を求める ことにした。
きわめて大きな非線形性を特性に持つ(摩擦力が関係するような)動作方程式に対してアナコンに よって一般的に取り扱うには， 先ず摩擦特性をどのよっに表示し， どのよ7な演算プログラムにする か考慮、しなければならない。 このような観点において， 筆者らは摩擦特性を表示する関数発生器， お よび演算精度の高い増幅器を試作した。 そしてこれを用いて， 先に説明した機械動作に相似なアナロ グ計算回路を組立てた。 この報告は， 二つの試作器の概要と その性能の実験結果を主に述べ， アナロ グ演算結果の一例を合わせて報告する。
2. 装 置 2.1 非線形関数発生器
関数形発生器は， 入力信号の大ききによって その特性曲線が原点を対称とする数本の折れ線として 得られるようにしたものである。 この折れ線は二個の不感帯特性， および二個の飽和特性を出力とす る演算器を組合せ， それぞれの演算器の出力を加算して求める関数形に近似した特性を得る方式で作 られている。 図1， および、図2は， この各要素
の接続の概念図， および精細な回路図である。
図2に於いて中央に示すピンコンタクトソケッ トは， 希望の非線形特性を得る以外に単独の不 感帯や飽和特性を得ることができるように， こ のソケット上で回路聞の接続を行うためのもの である。 そして， ポテンショメーターを調節し て， 各折れ点， および折れ線の勾配に変化を与 えることができる。 図3， および図4は各調節
各京都大学工学部
図l 概 念 図
40